A049600 -id:A049600 - cp's OEIS frontend

A049602 a(n) = (Fibonacci(2n)-(-1)^nFibonacci(n))/2.

0, 1, 1, 5, 9, 30, 68, 195, 483, 1309, 3355, 8900, 23112, 60813, 158717, 416325, 1088661, 2852242, 7463884, 19546175, 51163695, 133962621, 350695511, 918170280, 2403740304, 6293172025, 16475579353, 43133883845, 112925557953

Offset: 0

Clark Kimberling

A049602 gives the coefficients of x in the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1. For the constant terms, see A192352. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. - Clark Kimberling, Jun 29 2011

Index entries for linear recurrences with constant coefficients, signature (2,3,-4,1).

Cf. A049601.

LinearRecurrence[{2,3,-4,1},{0,1,1,5},30] (* Harvey P. Dale, Jul 07 2017 *)

a(n)=(fibonacci(2*n)-(-1)^n*fibonacci(n))/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n)=Sum{T(2i+1, n-2i-1): i=0, 1, ..., [ (n+1)/2 ]}, array T as in A049600.
Cosh transform of Fibonacci numbers A000045 (or mean of binomial and inverse binomial transforms of A000045). E.g.f.: cosh(x)(2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2). - Paul Barry, May 10 2003
a(n)=sum{k=0..floor(n/2), C(n, 2k)Fib(n-2k)}; - Paul Barry, May 01 2005
a(n)=2a(n-1)+3a(n-2)-4a(n-3)+a(n-4). - Paul Curtz, Jun 16 2008
G.f.: x(1-x)/((1+x-x^2)(1-3x+x^2)); a(n)=sum{k=0..n-1, (-1)^(n-k+1)*F(2k+2)*F(n-k+1)}; - Paul Barry, Jul 11 2008

Simpler description from Vladeta Jovovic and Thomas Baruchel, Aug 24 2004

More terms from Paul Curtz, Jun 16 2008

A049704 Array T read by antidiagonals; T(i,j)=number of nonnegative slopes of lines determined by two points in the triangular set {(x,y): 0<=x<=i, 0<=y<=j-j*x/i} of lattice points.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 4, 3, 1, 1, 1, 1, 3, 4, 4, 3, 1, 1, 1, 1, 4, 5, 6, 5, 4, 1, 1, 1, 1, 4, 6, 6, 6, 6, 4, 1, 1, 1, 1, 5, 6, 8, 10, 8, 6, 5, 1, 1, 1, 1, 5, 7, 9, 10, 10, 9, 7, 5, 1, 1, 1, 1, 6, 9, 11, 11, 12, 11, 11

Offset: 0

Clark Kimberling

			The array begins:
0 1 1 1 1 1 1 1 1...
1 1 1 1 1 1 1 1 1...
1 1 2 2 3 3 4 4 5...
1 1 2 4 4 5 6 6 7...
1 1 3 4 6 6 8 9 11...
...

Cf. A049600, A049615, A049639, A049687, A049695, A049702. Cf. A049706-A049709. Main diagonal: A002088.

t[i_,j_] := If[i==0||j==0, 1-KroneckerDelta[i+j], 1+Length[Union[Divide@@#& /@ Select[-Subtract@@@Subsets[Flatten[Table[{x,y}, {x,0,i}, {y,0,j-j*x/i}], 1], {2}], And@@Positive/@#&]]]];
(*Table[t[i,j], {i,0,10}, {j,0,10}]//TableForm*)
Flatten@Table[t[j,i-j], {i,0,20}, {j,0,i}]
(* Andrey Zabolotskiy, Jun 09 2017 *)

Name corrected by Michel Marcus and Andrey Zabolotskiy, Jun 10 2017

A055589 Convolution of A049612 with A011782.

Original entry on oeis.org

0, 1, 6, 26, 96, 321, 1002, 2972, 8472, 23392, 62912, 165504, 427264, 1085184, 2717184, 6718464, 16427008, 39763968, 95387648, 226951168, 535953408, 1257046016, 2929852416, 6789267456, 15648423936, 35888562176, 81927340032

Offset: 0

Wolfdieter Lang May 30 2000

Sixth column of triangle A055587. T(n,4) of array T as in A049600.

Index entries for linear recurrences with constant coefficients, signature (10, -40, 80, -80, 32).

Cf. A049600, A049612, A055587, A011782.

a(n)= T(n, 4)= A055587(n+4, 5).

G.f.: x*((1-x)^4)/(1-2*x)^5.

A055853 Convolution of A055852 with A011782.

Original entry on oeis.org

0, 1, 8, 43, 190, 743, 2668, 8989, 28814, 88720, 264224, 765088, 2162624, 5986304, 16268800, 43499264, 114629120, 298147840, 766361600, 1948794880, 4907171840, 12245598208, 30305419264, 74425892864, 181481635840, 439603953664

Offset: 0

Wolfdieter Lang May 30 2000

Eighth column of triangle A055587.

T(n,6) of array T as in A049600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Cf. A011782, A049600, A055587, A055852.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x)^6/(1-2*x)^7 )); // G. C. Greubel, Jan 16 2020

seq(coeff(series(x*(1-x)^6/(1-2*x)^7, x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 16 2020

CoefficientList[Series[x*(1-x)^6/(1-2*x)^7, {x,0,30}], x] (* G. C. Greubel, Jan 16 2020 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x)^6/(1-2*x)^7)) \\ G. C. Greubel, Jan 16 2020

def A055853_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)^6/(1-2*x)^7 ).list()
A055853_list(30) # G. C. Greubel, Jan 16 2020

a(n) = T(n, 6)= A055587(n+6, 7).

G.f.: x*(1-x)^6/(1-2*x)^7.

A035028 First differences of A002002.

Original entry on oeis.org

4, 20, 104, 552, 2972, 16172, 88720, 489872, 2719028, 15157188, 84799992, 475894200, 2677788492, 15102309468, 85347160608, 483183316512, 2739851422820, 15558315261812, 88462135512712, 503569008273992, 2869602773253884, 16368396446913420, 93449566652932784

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001850, A002002, A035029, A049600.

I:=[4,20,104]; [n le 3 select I[n] else ( (7*n+1)*Self(n-1) - (7*n-15)*Self(n-2) + (n-3)*Self(n-3) )/(n+1): n in [1..30]]; // G. C. Greubel, Oct 19 2022

Differences[CoefficientList[Series[((1-x)/Sqrt[1-6x+x^2]-1)/2, {x,0,30}], x]] (* Harvey P. Dale, Jun 04 2011 *)
With[{P=LegendreP}, Table[(n*(n+2)*P[n+2,3] -(6*n^2+10*n+1)*P[n+1,3] +(n+1)*(5*n+ 3)*P[n,3])/(2*n*(n+1)), {n,30}]] (* G. C. Greubel, Oct 19 2022 *)

def A001850(n): return gen_legendre_P(n,0,3)
def A035028(n): return ((n+1)*(n+3)*A001850(n+3) - (6*n^2 +22*n +17)*A001850(n+2) + (n+2)*(5*n+8)*A001850(n+1))/(2*(n+1)*(n+2))
[A035028(n) for n in range(40)] # G. C. Greubel, Oct 19 2022

a(n) = A049600(n, n-3).
D-finite with recurrence: (n+2)*a(n) - (7*n+8)*a(n-1) + (7*n-8)*a(n-2) - (n-2)*a(n-3) = 0. - R. J. Mathar, Jan 28 2020
a(n) = ((n+1)*(n+3)*A001850(n+3) - (6*n^2 +22*n +17)*A001850(n+2) + (n+2)*(5*n+8)*A001850(n+1))/(2*(n+1)*(n+2)), A001850(n) = LegrndreP(n, 3). - G. C. Greubel, Oct 19 2022

More terms from Harvey P. Dale, Jun 04 2011

A055854 Convolution of A055853 with A011782.

Original entry on oeis.org

0, 1, 9, 53, 253, 1059, 4043, 14407, 48639, 157184, 489872, 1480608, 4358752, 12541184, 35364864, 97960192, 267050240, 717619200, 1903452160, 4989337600, 12937052160, 33212530688, 84484882432, 213090238464, 533236219904

Offset: 0

Wolfdieter Lang May 30 2000

Ninth column of triangle A055587.

T(n,7) of array T as in A049600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Cf. A011782, A049600, A055587, A055853.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x)^7/(1-2*x)^8 )); // G. C. Greubel, Jan 16 2020

seq(coeff(series(x*(1-x)^7/(1-2*x)^8, x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 16 2020

CoefficientList[Series[x*(1-x)^7/(1-2*x)^8, {x,0,30}], x] (* G. C. Greubel, Jan 16 2020 *)
LinearRecurrence[{16,-112,448,-1120,1792,-1792,1024,-256},{0,1,9,53,253,1059,4043,14407,48639,157184},40] (* Harvey P. Dale, Nov 04 2023 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x)^7/(1-2*x)^8)) \\ G. C. Greubel, Jan 16 2020

def A055854_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)^7/(1-2*x)^8 ).list()
A055854_list(30) # G. C. Greubel, Jan 16 2020

a(n)= T(n, 7)= A055587(n+7, 8).

G.f.: x*(1-x)^7/(1-2*x)^8.

A100631 Triangle read by rows: T(n,k) = 2*(T(n-1,k-1) - T(n-2,k-1) + T(n-1,k)) for 0 < k < n, T(n,0) = T(n,n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 12, 8, 1, 1, 16, 32, 32, 16, 1, 1, 32, 80, 104, 80, 32, 1, 1, 64, 192, 304, 304, 192, 64, 1, 1, 128, 448, 832, 1008, 832, 448, 128, 1, 1, 256, 1024, 2176, 3072, 3072, 2176, 1024, 256, 1, 1, 512, 2304, 5504, 8832, 10272, 8832, 5504, 2304, 512, 1

Offset: 0

Reinhard Zumkeller, Dec 03 2004

From Petros Hadjicostas, Feb 09 2021: (Start)
The rectangular version (R(n,k): n,k >= 0) of this symmetric triangular array (T(n,k): 0 <= k <= n) is given by R(n,k) = T(n+k,k) for n,k >= 0. Conversely, T(n,k) = R(n-k, k) for 0 <= k <= n.
Note that [o.g.f of R](x,y) = [o.g.f. of T](x, y/x) and [o.g.f of T](x,y) = [o.g.f of R](x,x*y). (End)
From Petros Hadjicostas, Feb 10 2021: (Start)
All the conjectures below are true because one has to prove only one of them, and the rest follow from the proved one.
As Peter Luschny pointed out, one has to show only that the function S(n,k) = 2^n*hypergeom([-k + 1, n], [1], -1) satisfies the recurrence S(n,k) = 2*(S(n,k-1) - S(n-1,k-1) + S(n-1,k)) for n, k > 0 and the initial conditions S(n,0) = S(0,n) = 1 for n >= 0.
This is quite easy to achieve because S(n,k) = 2^n*Sum_{s=0}^{k-1} binomial(k-1,s)*binomial(n+s-1,s) for n >= 0 and k >= 1. The proof of the recurrence relies on the identity binomial(m,n) = binomial(m-1, n) + binomial(m-1,n-1).
Note that without the 2^n in the formula R(n,k) = 2^n*hypergeom([-k + 1, n], [1], -1), we essentially get array A049600.
In addition, note that without the 2^(n-k-1) in the formula T(n,k+1) = 2^(n-k-1)*hypergeom([-k, n-k+1], [1], -1), we essentially get A208341 (without the first column and the main diagonal of T). (End)

			From _Petros Hadjicostas_, Feb 09 2021: (Start)
Triangle T(n,k) (with rows n >= 0 and columns 0 <= k <= n) begins:
  1,
  1,   1,
  1,   2,    1,
  1,   4,    4,    1,
  1,   8,   12,    8,    1,
  1,  16,   32,   32,   16,    1,
  1,  32,   80,  104,   80,   32,    1,
  1,  64,  192,  304,  304,  192,   64,    1,
  1, 128,  448,  832, 1008,  832,  448,  128,   1,
  1, 256, 1024, 2176, 3072, 3072, 2176, 1024, 256, 1,
  ...
Rectangular array R(n,k) (with rows n >= 0 and columns k >= 0) begins:
  1,   1,    1,    1,     1,     1,      1,       1, ...
  1,   2,    4,    8,    16,    32,     64,     128, ...
  1,   4,   12,   32,    80,   192,    448,    1024, ...
  1,   8,   32,  104,   304,   832,   2176,    5504, ...
  1,  16,   80,  304,  1008,  3072,   8832,   24320, ...
  1,  32,  192,  832,  3072, 10272,  32064,   95104, ...
  1,  64,  448, 2176,  8832, 32064, 107712,  341504, ...
  1, 128, 1024, 5504, 24320, 95104, 341504, 1150592, ...
  ... (End)

Index entries for triangles and arrays related to Pascal's triangle

Cf. A000079, A001787, A049600, A084773, A087161, A100312, A152254, A208341, A341344.

T[, 0] = T[n, n_] = 1;
T[n_, k_] /; 0, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 03 2021 *)

From Petros Hadjicostas, Feb 09 2021: (Start)
Formulas for the triangular array (T(n,k): 0 <= k <= n):
T(n,k) = T(n,n-k) for 0 <= k <= n.
Sum_{k=0..n} T(n,k) = A087161(n+1).
T(n,1) = T(n,n-1) = 2^(n-1) = A000079(n-1) for n >= 1.
T(n,2) = T(n,n-2) = (n-1)*2^(n-2) = A001787(n-1) for n >= 2.
T(n,3) = T(n,n-3) = (n^2-n-4)*2^(n-4) = A100312(n-3) for n >= 3.
T(n,floor(n/2)) = T(n,ceiling(n/2)) = A341344(n).
Bivariate o.g.f.: Sum_{n,k >= 0} T(n,k)*x^n*y^k = (3*x^2*y - 2*x*y - 2*x + 1)/((1 - x)*(-x*y + 1)*(2*x^2*y - 2*x*y - 2*x + 1)).
Conjecture based on Peter Luschny's formulas in other sequences: T(n,k) = 2^(n-k)*hypergeom([-k + 1, n-k], [1], -1) = 2^k*hypergeom([-(n-k) + 1, k], [1], -1).
Formulas for the rectangular array (R(n,k): n,k >= 0):
R(n,k) = 2*(R(n,k-1) - R(n-1,k-1) + R(n-1,k)) for n,k > 0 with R(n,0) = R(0,n) = 1 for n >= 0.
R(n,k) = R(k, n) for n,k >= 0.
R(1,n) = R(n,1) = 2^n = A000079(n).
R(2,n) = R(n,2) = (n+1)*2^n = A001787(n+1).
R(3,n) = R(n,3) = (n^2+5*n+2)*2^(n-1) = A100312(n).
R(n,n) = A152254(n-1) = 2*A084773(n-1) for n >= 1.
Bivariate o.g.f.: Sum_{n,k >= 0} R(n,k)*x^n*y^k = (3*x*y - 2*x - 2*y - 1)/((1 - x)*(1 - y)*(2*x*y - 2*x - 2*y - 1)).
Conjecture based on Peter Luschny's formulas in other sequences: R(n,k) = 2^n*hypergeom([-k + 1, n], [1], -1) = 2^k*hypergeom([-n + 1, k], [1], -1). (End)
From Petros Hadjicostas, Feb 10 2021: (Start)
The above conjecture is true (see the comments).
R(n,k) = 2^k*Sum_{s=0}^{n-1} binomial(n-1,s)*binomial(k+s-1,s) = 2^n*Sum_{s=0}^{k-1} binomial(k-1,s)*binomial(n+s-1,s) for n, k >= 1.
To get two binomial formulas for T(n,k), use the equation T(n,k) = R(n-k, k) for 1 <= k <= n and the above formulas for R(n,k). (End)

Offset changed by Petros Hadjicostas, Feb 09 2021

A055855 Convolution of A055854 with A011782.

Original entry on oeis.org

0, 1, 10, 64, 328, 1462, 5908, 22180, 78592, 265729, 864146, 2719028, 8316200, 24814832, 72453344, 207502016, 584094080, 1618757120, 4423347200, 11932579840, 31812874240, 83901227008, 219074805760, 566754967552

Offset: 0

Wolfdieter Lang May 30 2000

Tenth column of triangle A055587.

T(n,8) of array T as in A049600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-144,672,-2016,4032,-5376,4608,-2304,512).

Cf. A011782, A049600, A055584, A055857.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x)^8/(1-2*x)^9 )); // G. C. Greubel, Jan 16 2020

seq(coeff(series(x*(1-x)^8/(1-2*x)^9, x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 16 2020

CoefficientList[Series[x*(1-x)^8/(1-2*x)^9, {x,0,30}], x] (* G. C. Greubel, Jan 16 2020 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x)^8/(1-2*x)^9)) \\ G. C. Greubel, Jan 16 2020

def A055855_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)^8/(1-2*x)^9 ).list()
A055855_list(30) # G. C. Greubel, Jan 16 2020

a(n) = T(n, 8) = A055587(n+8, 9).

G.f.: x*(1-x)^8/(1-2*x)^9.

cp's OEIS Frontend

A049602 a(n) = (Fibonacci(2*n)-(-1)^n*Fibonacci(n))/2.

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A049704 Array T read by antidiagonals; T(i,j)=number of nonnegative slopes of lines determined by two points in the triangular set {(x,y): 0<=x<=i, 0<=y<=j-j*x/i} of lattice points.

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A055589 Convolution of A049612 with A011782.

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A055853 Convolution of A055852 with A011782.

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A035028 First differences of A002002.

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A055854 Convolution of A055853 with A011782.

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A100631 Triangle read by rows: T(n,k) = 2*(T(n-1,k-1) - T(n-2,k-1) + T(n-1,k)) for 0 < k < n, T(n,0) = T(n,n) = 1.

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A049602 a(n) = (Fibonacci(2n)-(-1)^nFibonacci(n))/2.